Section 13.9: Exploring Solutions to the Coulomb Problem

Animation 1: Energy Eigenfunction Amplitude  Animation 2: Energy Eigenfunction Amplitude and Pieces
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The entire solution to the Coulomb problem can be represented as
ψ_{nlm} = A_{nl }R_{nl}(r) Y_{l}^{m}(θ,φ) , (13.56)
where A_{nl} are the normalization constants, R_{nl}(r) are the radial wave functions, and Y_{l}^{m}(θ,φ) are the spherical harmonics.
In this Exploration, Animation 1 depicts ψ_{nlm} in the zx plane only. In Animation 2, Φ_{m}(φ), P_{l}^{m} (θ), R(r), and ψ_{nlm} are visualized in one of four panels. The entire solution, ψ_{nlm}, is visualized in the zx plane only. To generate the spherical energy eigenfunction, first imagine the rotation of ψ_{nlm} about the z axis; this gives you the shape of the energy eigenfunction. Then, to get the phase of the energy eigenfunction, project the phase (color) from Φ_{m}(φ).
 For a given value of n and l, how does the number of angular lobes in P_{l}^{m} (θ) change with m?
 For a given value of n and l, how does the number of wavelengths (from blue to blue is one wavelength) in Φ_{m}(φ) change with m?
 How do the nonzero l values affect the radial energy eigenfunction?
 How do the nonzero m values affect the radial energy eigenfunction?
 For n = 3, how many times does the radial energy eigenfunction cross zero (change signs) for each possible value of l? Try this for a few other values for the principal quantum number, n, and see if your conclusion holds.
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